11 Responses to Work 1

A two-level formal language consists of a set of “words” and a set of “sentences”, where a word is defined as a finite sequence of symbols from a finite alphabet, and a sentence is defined as a finite sequence of words. The finite-state structure of a two-level formal language can be captured in a couple of finite-state transition trees, one for the words and one for the sentences. Recording data about the frequencies of node traversals allows us to compute relative probabilities and local entropies and many other statistics of interest, and the resulting structures can be used to predict likely completions of word and sentence fragments.

I spent a couple of my parallel lives in the 1980s developing an AI sort of program that combined faculties for data-driven empirical learning with faculties for concept-driven logical modeling. One of the things I learned along the way is that there appears to be a trade-off between these two modes of processing that makes it very difficult to integrate empiricist and rationalist faculties within a single intelligent agent. Explains a lot about the history of thought, I think.

I’ve always liked the phonetic links between the Latin forma (form, beauty), the English formidable, and the French formidable, the last two going back to Latin formido (fear) and Greek mormō (she-monster).

The number of students served by a community school is a handy “rule of thumb” for assessing the monetary need. It is also reasonable to support the public resource that meets that need according to fair measures of ability to pay and benefit received.

That is the rationale of public support for public education to which reasonable people have consented through all the glory days of U.S. history. It is a holistic way of looking at educational systems as complex, dynamic, interrelated wholes, involving the entire community of parents, students, teachers, and policymakers. The atomistic way of looking at a living system, where individuals vie to tear off their individual bits of flesh and drain their individual portion of “stone soup” down to the last drop — that is a species of society that will rapidly dry up and die. Let’s hope we don’t have to prove that in practice, since so many societies have proven it already in the past.

There are people who say that policy decisions about the character and conduct of our schools should be decided by a system where parents and students are given the franchise to “vote with their feet”. That is absurd. At no point does the remainder of the citizenry delegate their duties and powers in the matter of education to parents of school-age children according to the number of children they have in school for the quickly passing time they are there. It is a patent violation of democratic principles of representation to dictate that parents alone should have the power to sell off property that belongs to all and to liquidate resources that the long generations before us have entrusted to the future of us all.

One of the things Boole sought to do in his Laws of Thought was to build a bridge between logic and probability theory. That is no small task if one considers the gulf between rationalist and empiricist tendencies of mind, or what translates into computational terms as the problem of integrating concept-driven and data-driven algorithms for computational learning and reasoning.

At any rate, the language we use in probability and statistics is very apt when it comes to describing Boolean functions. There we call a function an indicator function.

Passage from arbitrary to cell map …

The diagram shows the factoring of the proposition as the functional composition , that is to say, where is the coding of each as an k-bit string in and where is the mapping of codes into a co-domain that we interpret as logical values,

In this picture is just the rectangular area of an ordinary Venn diagram, a universe of discourse that can be filled with whatever we choose. And is just the shading in of a part of its area, where the functional value indicates the elements that it means to indicate.

Now, what will quickly develop from this picture is that a person will soon pick out a finite number of his or her favorite propositions. These propositions are optimally chosen to be independent of each other, that is, orthogonal in a logical sense, and are commonly dubbed as one’s basic propositions or singled out by referring to them as coordinate projections of the form for I usually picture these as the “circles” of the Venn diagram. After that, if a given system of basic propositions is moderately adequate to the task of describing, more or less approximately, every other area of arbitrary shape that one needs to cover in the universe of discourse then one will find it convenient to factor any cell map through the cartesian product as in the following diagram:

This says that where we can think of the bit-list as the binary coding of the element and where is the derived mapping from codes to

Given this sort of set-up, we can proceed to work with derived propositions using truth tables or something equivalent.

There are actually several types of Boolean functions that are represented in a typical Venn diagram. They all have the Boolean domain or one of its powers as their functional codomains but their functional domains may vary, not all being limited to finite cardinalities. As an aid to sorting out their variety, consider the array of functional arrows in the following figure.

Suppose is a universe of discourse represented by the rectangular area of a Venn diagram. Note that the set itself may have any cardinality. The most general type of Boolean function is a map This is known as a Boolean-valued function since only its functional values need be in

A function of the type is called a characteristic function in set theory or an indicator function in probability and statistics since it characterizes or indicates a particular subset of , namely, the so-called fiber or inverse image of the value for which we have the notation and definition

The notation is often used for the characteristic function of a subset of All together then, we have